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Properties

Label 2-72-72.59-c3-0-16
Degree $2$
Conductor $72$
Sign $0.998 - 0.0552i$
Analytic cond. $4.24813$
Root an. cond. $2.06110$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.80 + 2.17i)2-s + (−3.91 + 3.42i)3-s + (−1.46 − 7.86i)4-s + (7.74 − 13.4i)5-s + (−0.371 − 14.6i)6-s + (−4.08 + 2.35i)7-s + (19.7 + 11.0i)8-s + (3.59 − 26.7i)9-s + (15.1 + 41.1i)10-s + (4.32 − 2.49i)11-s + (32.6 + 25.7i)12-s + (66.4 + 38.3i)13-s + (2.25 − 13.1i)14-s + (15.6 + 79.0i)15-s + (−59.7 + 23.0i)16-s − 120. i·17-s + ⋯
L(s)  = 1  + (−0.639 + 0.769i)2-s + (−0.752 + 0.658i)3-s + (−0.182 − 0.983i)4-s + (0.693 − 1.20i)5-s + (−0.0252 − 0.999i)6-s + (−0.220 + 0.127i)7-s + (0.873 + 0.487i)8-s + (0.132 − 0.991i)9-s + (0.480 + 1.30i)10-s + (0.118 − 0.0685i)11-s + (0.784 + 0.619i)12-s + (1.41 + 0.818i)13-s + (0.0430 − 0.250i)14-s + (0.268 + 1.36i)15-s + (−0.933 + 0.359i)16-s − 1.72i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0552i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0552i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $0.998 - 0.0552i$
Analytic conductor: \(4.24813\)
Root analytic conductor: \(2.06110\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{72} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 72,\ (\ :3/2),\ 0.998 - 0.0552i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.902292 + 0.0249448i\)
\(L(\frac12)\) \(\approx\) \(0.902292 + 0.0249448i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.80 - 2.17i)T \)
3 \( 1 + (3.91 - 3.42i)T \)
good5 \( 1 + (-7.74 + 13.4i)T + (-62.5 - 108. i)T^{2} \)
7 \( 1 + (4.08 - 2.35i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (-4.32 + 2.49i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-66.4 - 38.3i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 120. iT - 4.91e3T^{2} \)
19 \( 1 - 30.8T + 6.85e3T^{2} \)
23 \( 1 + (-91.9 + 159. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-28.8 - 49.9i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (37.8 + 21.8i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 108. iT - 5.06e4T^{2} \)
41 \( 1 + (230. + 133. i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (105. + 181. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-191. - 332. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 419.T + 1.48e5T^{2} \)
59 \( 1 + (17.1 + 9.91i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-158. + 91.4i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (215. - 373. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 772.T + 3.57e5T^{2} \)
73 \( 1 + 433.T + 3.89e5T^{2} \)
79 \( 1 + (-644. + 372. i)T + (2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (408. - 235. i)T + (2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 206. iT - 7.04e5T^{2} \)
97 \( 1 + (-713. - 1.23e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.27759214656699717062522777429, −13.22347656165244026723334348791, −11.74730571715629264487219767236, −10.53690067021129966330984352147, −9.220936306144582194223931772680, −8.894460087751121125891410633770, −6.77909355345201367566808049869, −5.63777184837284400118144680563, −4.63481230646425496867790029900, −0.920384015480600605360090159118, 1.53524323135448398526695052450, 3.35122073978563504164958518563, 5.90581879505814656616080057838, 7.01583587799939749787375844818, 8.338657996349754676127543698095, 10.09691073964804319866231829844, 10.73761277252149698920135828252, 11.62180636349010140816520465424, 13.07064209235846150379263086051, 13.55295952558952452701592322398

Graph of the $Z$-function along the critical line